Abstract

We study the equilibrium of a liquid film on an attractive spherical substrate for an intermolecular interaction
model exhibiting both fluid-fluid and fluid-wall long-range forces. We first reexamine the wetting properties of
the model in the zero-curvature limit, i.e., for a planar wall, using an effective interfacial Hamiltonian approach
in the framework of the well known sharp-kink approximation (SKA). We obtain very good agreement with
a mean-field density functional theory (DFT), fully justifying the use of SKA in this limit. We then turn our
attention to substrates of finite curvature and appropriately modify the so-called soft-interface approximation
(SIA) originally formulated by Napiórkowski and Dietrich [Phys. Rev. B 34, 6469 (1986)] for critical wetting
on a planar wall. A detailed asymptotic analysis of SIA confirms the SKA functional form for the film growth.
However, it turns out that the agreement between SKA and our DFT is only qualitative. We then show that the
quantitative discrepancy between the two is due to the overestimation of the liquid-gas surface tension within
SKA. On the other hand, by relaxing the assumption of a sharp interface, with, e.g., a simple ÒsmoothingÓ of
the density profile there, markedly improves the predictive capability of the theory, making it quantitative and
showing that the liquid-gas surface tension plays a crucial role when describing wetting on a curved substrate.
In addition, we show that in contrast to SKA, SIA predicts the expected mean-field critical exponent of the
liquid-gas surface tension.

pacs:

The behavior of fluids in confined geometries, in particular in the
vicinity of solid substrates, and associated wetting phenomena are
of paramount significance in numerous technological applications and
natural phenomena. Wetting is also central in several fields, from
engineering and materials science to chemistry and biology. As a
consequence, it has received considerable attention, both
experimentally and theoretically for several decades. Detailed and
comprehensive reviews are given in
Refs. Dietrich (1988); Bonn et al. (2009); Schick (1990); Sullivan and da Gama (1986).

Once a substrate (e.g. a solid wall) is brought into contact with a gas, the substrate-fluid attractive forces cause adsorption of some of the fluid molecules on the substrate surface, such that at least a microscopically thin
liquid film forms on the surface. The interplay between the fluid-fluid interaction (cohesion) and the fluid-wall interaction (adhesion) then determines a particular wetting state of the system. This state can be quantified by
the contact angle at which the liquid-gas interface meets the substrate. If the contact angle is non-zero, i.e. a spherical cap of the liquid is formed on the substrate, the surface is called partially wet. In the regime of
partial wetting, the cap is surrounded by a thin layer of adsorbed fluid which is of molecular dimension. Upon approaching the critical temperature, the contact angle continuously decreases and eventually vanishes. Beyond this
wetting temperature one speaks of complete wetting and the film thickness becomes of macroscopic dimension. The transition between the two regimes can be qualitatively distinguished by the rate of the disappearance of the
contact angle, which is discontinuous in the case of a first-order transition or continuous for critical wetting.

From a theoretical point of view, it is much more convenient to take
the adsorbed film thickness, ℓ, rather than the contact angle,
as an order parameter for wetting transitions and related phenomena. An
interfacial Hamiltonian is then minimized with respect to ℓ as
is typically the case with the (mesoscopic) Landau-type field
theories and (microscopic) density functional theory (DFT) – where
ℓ can be easily determined from the Gibbs adsorption, a direct
output of DFT.

In this study, we examine the wetting properties of a simple fluid
in contact with a spherical attractive wall by using an
intermolecular interaction model with fluid-fluid and fluid-wall
long-range forces. The curved geometry of the system prohibits a
macroscopic growth of the adsorbed layer (and thus complete
wetting), since the free energy contribution due to the liquid-gas
interface increases with the film thickness ℓ, and thus for a
given radius of a spherical substrate there must be a maximum finite
value of
ℓDietrich (1988); Bieker and Dietrich (1998); Hołyst and Poniewierski (1987).
For the mesoscopic approaches, the radius of the wall R, is a new
field variable that introduces one additional ℓ-dependent term
to the effective interface Hamiltonian of the system, compared to
the planar geometry, where the only ℓ-dependent term is the
binding potential between the wall-liquid and liquid-gas interfaces.
Furthermore, for a fluid model exhibiting a gas-liquid phase
transition, such as ours, it has been found that two regimes of the
interfacial behavior should be distinguished: R>RC, in which
case the surface tension can be expanded in integer powers of
R−1 and R<RC, where the interfacial quantities exhibit a
non-analytic behavior Evans et al. (2004). Moreover, for an
intermolecular interaction model with fluid-fluid long-range
interactions, there is an additional R−2logR contribution to
the surface tension in the R>RC regime Stewart and Evans (2005a). These
striking observations actually challenge all curvature expansion
approaches. In addition, a certain equivalence between a system of a
saturated fluid on a spherical wall and a system of an unsaturated
fluid on a planar wall above the wetting temperature has been
found Bieker and Dietrich (1998); Stewart and Evans (2005a).
Somewhat surprisingly, DFT computations confirmed this
correspondence at the level of the density profiles down to
unexpectedly small radii of the wall Stewart and Evans (2005a).

Most of these conjectures follow from the so-called sharp-kink
approximation (SKA) Dietrich (1988), based on a simple piece-wise
constant approximation of a one-body density distribution of the
fluid, i.e. a coarse-grained approach providing a link between
mesoscopic Hamiltonian theories and microscopic DFT. The simple
mathematical form of SKA has motivated many theoretical
investigations of wetting phenomena as it makes them analytically
tractable. At the same time SKA appears to capture much of the
underlying fundamental physics for planar substrates (often in
conjugation with exact statistical mechanical sum
rules Henderson (1992)).

However, as we show in this work, SKA is only qualitative
for spherical substrates, even though the functional form of the
film growth can still be successfully inferred from the
theory Stewart and Evans (2005a).
We attribute this to the particular approximation of the liquid-gas
interface adapted by SKA. In particular, since the ℓ-dependent
contribution to the interface Hamiltonian due to the curvature is
proportional to the liquid-gas surface tension, the latter plays an
important role compared to the planar geometry.

More
specifically, the curved geometry induces a Laplace pressure whose
value depends on both film thickness and the surface tension and so
the two quantities are now coupled, in contrast with the planar
geometry where a parallel shift of the liquid-gas dividing surface
does not influence the surface contribution to the free energy of
the system.
We further employ an alternative coarse-grained approach, a
modification of the one originally proposed by Napiórkowski and
Dietrich Napiórkowski and Dietrich (1986) for the planar
geometry, which replaces the jump in the density profile at the
liquid-gas interface of SKA by a continuous function restricted by
several reasonable constraints. We show that in this
“soft-interface approximation” (SIA) the leading curvature
correction to the liquid-gas surface tension is O(R−1), rather
than O(R−2logR), in line with the Tolman theory. Once a
particular approximation for the liquid-gas interface is taken, the
corresponding Tolman length can be easily determined. Apart from
this, we find that the finite width of the liquid-gas interface
significantly improves the prediction of the corresponding surface
tension when compared with the microscopic DFT computations, which
consequently markedly improves the estimation of the film thickness
in a spherical geometry.

In Sec. II we describe our microscopic model and the
corresponding DFT formalism. In
Sec. III we present results of
wetting phenomena on a planar wall obtained from our DFT based on a
continuation scheme that allows us to trace metastable and unstable
solutions. The results are compared with the analytical prediction
as given by a minimization of the interface Hamiltonian based on
SKA. We also make a connection between the two approaches by
introducing the microscopic model into the interfacial Hamiltonian.
In Sec. IV we turn our attention to
the main part of our study, a thin liquid film on a spherical wall.
We show that the SKA does not perform as well as might be desired, in
particular, it does not account for a quantitative description of
the liquid-gas surface tension which plays a significant role when
the substrate geometry is curved. We then introduce SIA and present
an asymptotic analysis with the new approach. Comparison with DFT
computations reveals a substantial improvement of the resulting
interface Hamiltonian even for very simple approximations of the
density distribution at the liquid-vapour interface, indicating the
significance of a non-zero width of the interface. We conclude in
Sec. V with a summary of our results and discussion.
Appendix A describes the continuation method we developed for the
numerical solution of DFT. In Appendix B we show derivations of the
surface tension and the binding potential for both a planar and a
spherical geometry within SKA. Finally, Appendix C shows derivations
of the above quantities, including Tolman’s length, using SIA.

ii.1 General formalism

DFT is based on Mermin’s proof Mermin (1965) that the free
energy of an inhomogeneous system at equilibrium can be expressed as
a functional of an ensemble averaged one-body density, ρ(r) (see
e.g. Ref. Evans (1979) for more details). Thus, the free-energy
functional F[ρ] contains all the equilibrium physics
of the system under consideration. Clearly, for a 3D fluid model one
has to resort to an approximative functional. Here we adopt a simple
but rather well established local density approximation

F[ρ]=

∫fHS(ρ(r))ρ(r)dr+

+12∬ρ(r)ρ(r′)ϕ(|r−r′|)dr′dr,

(1)

where fHS(ρ(r)) is the free energy per particle of
the hard-sphere fluid (accurately described by the Carnahan-Starling
equation of state), including the ideal gas contribution. The
contribution due to the long-range van der Waals forces is included
in the mean-field manner. To be specific, we consider a full
Lennard-Jones 12-6 (LJ) potential to model the fluid-fluid
attraction according to the Barker-Henderson perturbative scheme

ϕ(r)={0r<σ4ε((σr)12−(σr)6)r≥σ,

(2)

where for the sake of simplicity the Lennard-Jones parameter σ is taken equal to the hard-sphere diameter.

The free-energy functional, F[ρ], describes the
intrinsic properties of a given fluid. The total free energy
including also a contribution of the external field is related to
the grand potential functional through the Legendre transform

Ω[ρ]=F[ρ]+∫ρ(r)(V(r)−μ)dr,

(3)

where μ is the chemical potential and V(r) is the external field due to the presence of a wall W⊂R3,

V(r)={∞r∈Wρw∫Wϕw(|r−r′|)dr′ elsewhere,

(4)

consisting of the atoms interacting with the fluid particles via the
Lennard-Jones potential, ϕw(r), with the parameters
σw and εw, and uniformly distributed throughout
the wall with a density ρw:

ϕw(r)=4εw((σwr)12−(σwr)6).

(5)

Applying the variational principle to the grand potential
functional, Eq. (3), we attain the
Euler-Lagrange equation:

δFHS[ρ]δρ(r)+∫ρ(r′)ϕ(|r−r′|)dr′+V(r)−μ=0,

(6)

where FHS[ρ] denotes the first term in the
right-hand-side of (1). In general, the
solution of (6) comprises all extremes of
the grand potential
Ω[ρ] as given by (3) and
not just the global minimum corresponding to the equilibrium state.
Here we develop a
pseudo arc-length continuation scheme for
the numerical computation of (6) that
enables us to capture both locally stable and unstable solutions and
thus to construct the entire bifurcation diagrams for the isotherms
(details of the scheme are given in Appendix
A).

The excess part of the grand potential functional
(3) over the bulk may be expressed in the
form

Ωex[ρ(r)]=

−∫(p(ρ(r))−p(ρb))dr+

+

12∬ρ(r)(ρ(r′)−ρ(r))ϕ(|r′−r|)dr′dr+

+

∫ρ(r)V(r)dr,

(7)

where ρb is the density of the bulk phase and

−p(ρ)=ρfHS(ρ)+αρ2−μρ,

(8)

is the negative pressure, or grand potential per unit volume, of a
system with uniform density ρ and α≡12∫ϕ(|r|)dr=−169πεσ3. In particular, the equilibrium value of the
excess grand potential (7) per
unit area of a two-phase system of liquid and vapour in the absence
of an external field, yields the surface tension between the
coexisting phases, γlg. The prediction of γlg as
given by minimization of (7)
agrees fairly well with both computations and experimental data as
shown in Fig. 1.

ii.2 Translational symmetry: planar wall

If the general formalism outlined above is applied on a particular
external field attaining a certain symmetry, it will adopt a
significantly simpler form. In the next subsection we will formulate
the basic equations resulting from the equilibrium conditions
obtained from the minimization
of (7), for a spherical model
of the external field, i.e. a system with rotational symmetry. But
prior to that, it is instructive to discuss the zero-curvature limit
of the above model, corresponding to an adsorbed LJ fluid on a
planar wall, a system with translational symmetry.

For a planar substrate W=R2×R− in
Cartesian coordinates, the density profile is only a function of
z, so that the Euler-Lagrange equation reads

μHS(ρ(z))+∫∞0ρ(z′)ΦPla(|z−z′|)dz′+

(9)

+V∞(z)−μ=0

(∀z∈R+),

where μHS(ρ)=∂(fHS(ρ)ρ)∂ρ is
the chemical potential of the hard-sphere system.
A fluid particle at a distance z from the wall experiences the
wall potential:

V∞(z)

=ρw∫Wϕw(√x′2+y′2+(z−z′)2)dx′dy′dz′

={∞z≤04πρwεwσ3w(145(σwz)9−16(σwz)3)z>0.

(10)

ΦPla(z) in Eq. (9) is the
surface potential exerted by the fluid particles uniformly
distributed (with a unit density) over the x-y plane at distance
z:

ΦPla(z)

=∬ϕ(√x2+y2+z2)dydx

=2π∫∞0ϕ(√z2+r2)rdr

(11)

=−65πεσ2×{ll1z<σ,53(σz)4−23(σz)10z≥σ.

In the framework of DFT, the natural order parameter for wetting
transitions is the Gibbs adsorption per unit area:

Γ∞[ρ(z)]=∫∞0(ρ(z)−ρb)dz.

(12)

ii.3 Rotational symmetry: spherical wall

If the external field is induced by a spherical wall, W={r∈R3:r≡|r|<R}, the variational principle
yields

μHS(ρ(r))+∫∞Rρ(r′)ΦSph(r,r′)dr′+

(13)

+VR(r)−μ

=0,(∀r>R),

where ΦSph(r,r′) is the surface interaction potential
per unit density generated by fluid particles uniformly distributed
on the surface of the sphere Br′ centered at the origin at
distance r,

ΦSph(r,r′)

=∫∂Br′ϕ(|r−~r|)d~r.

(14)

=r′r(ΦPla(|r−r′|)−ΦPla(|r+r′|))

(see also Appendix B.1). The wall
potential in Eq. (4) for the
spherical wall W={r∈R3:|r|≤R} is:

VR(r)=

ρwεwσ4wπ3r{σ8w30[r+9R(r+R)9−r−9R(r−R)9]+

+σ2w[r−3R(r−R)3−r+3R(r+R)3]}.

(15)

Replacing the distance from the origin r by the radial distance from the wall ~r=r−R, one can easily see
that the external potential (15)
reduces to the planar wall potential (10), for
R→∞.
Analogously to the planar case, we define the
adsorption ΓR as the excess number of particles of the
system with respect to the surface of the wall:

In this section we make a comparison between the numerical solution
of DFT and the prediction given by the effective interfacial
Hamiltonian according to SKA for the first-order wetting transition
on the planar substrate. We consider a planar semi-infinite wall
interacting with the fluid according to
(10) with the typical parameters
ρwεw=0.8ε/σ3 and σw=1.25σ that correspond to the class of intermediate-substrate
systems Pandit et al. (1982) for which prewetting phase
transitions can be observed. We note that wetting on planar and
spherical walls is a multiparametric problem and hence a full
parametric study of the global phase diagram is a difficult task,
beyond the scope of this paper.

iii.1 Numerical DFT results of wetting on a planar wall

Figure 2 depicts the surface-phase diagram of
the considered model in the (Δμ,T) plane, where Δμ=μ−μsat is the departure of the chemical potential from
its saturation value. The first-order wetting transition takes place
at wetting temperature kBTw=0.621ε, well bellow the
critical temperature of the bulk fluid kBTc=1.006ε
for our model. The prewetting line connects the saturation line at
the wetting temperature Tw and terminates at the prewetting
critical point, kBTpwc=0.724ε. The slope of the
prewetting line is governed by a Clapeyron-type
equation Hauge and Schick (1983), which, in particular, states that the
prewetting line approaches the saturation line tangentially at Tw
with

d(Δμpw)dT∣∣∣T=Tw=0,

(17)

in line with our numerical computations. Schick and
Taborek Schick and Taborek (1992) later showed that the prewetting line
scales as −Δμ∼(T−Tw)3/2. In
Ref. Bonn and Ross (2001), this power law was confirmed experimentally,
such that

−Δμpw(T)kBTw=C(T−TwTw)3/2,

(18)

with C≈12Bonn and Ross (2001). A fit of our DFT
results with (18) leads to a coefficient
C=0.77, in a reasonable agreement with the experimental data –
see Fig. 2.

Figure 2:
(a) The deviation of the chemical potential from its
saturation value at prewetting (crosses), and at the left (open squares)
and right (filled squares) saddle nodes of bifurcation as a
function of temperature. The dashed line marks the locus of the
chemical potential at saturation for the given temperature,
Δμ=0.
The solid line is a fit to −Δμpw(T)/(kBTw)=C((T−Tw)/Tw)3/2
where the wetting temperature is kBTw=0.621ε and the prewetting critical temperature is
kBTpwc=0.724ε. The resulting
coefficient is C=0.77.
(b) The scaled prewetting phase diagrams for different systems.
The circles are DFT calculations for an attractive wall with
σw=1.25σ and ρwεw=0.8ε/σ3 (open circles) and
ρwεw=0.75ε/σ3 (filled circles).
Experimental data Bonn and Ross (2001): filled squares, methanol on cyclohexane (Kellay
et. al. 1993) H.Kellay et al. (1992); open triangles, H2 on rubidium
(Mistura et al. 1994) Mistura et al. (1994);
filled triangles, He on caesium (Rutledge et al., 1997) Rutledge and Taborek (1992);
open squares, H2 on Caesium (Ross et al., 1997) Ross et al. (1997).

Figure 3 depicts the adsorption isotherm in
terms of the thickness of the adsorbed liquid film ℓ as a
function of Δμ for the temperature kBT=0.7ε
and in the interval between the wetting temperature Tw and the
prewetting critical temperature Tpwc. ℓ can be associated
with the Gibbs adsorption through

ℓ=ΓR[ρ]Δρ,

(19)

for both finite and infinite R, where
Δρ=ρsatl−ρsatg is the difference between
the liquid and gas densities at saturation.

The isotherm exhibits a van der Waals loop with two turning points
depicted as B and C demarcating the unstable branch. Points A and D
indicate the equilibrium between thin and thick layers,
corresponding to a point on the prewetting line in
Fig. 2. The location of the equilibrium points
can be obtained from a Maxwell construction. Details of the
numerical scheme we developed for tracing the adsorption isotherms
are given in Appendix A.

Figure 3:
(a) The ℓ–Δμ
bifurcation diagram for kBT=0.7ε for a wall with
ρwεw=0.8ε/σ3 and σw=1.25σ.
Δμ is the deviation of the chemical potential from its
saturation value, μsat. The prewetting transition, marked by
the dashed line, occurs at chemical potential Δμpw=−0.022ε.
The inset subplots show the density ρσ3 as a function of
the distance z/σ from the wall. (b) The excess grand potential
Ωex/ε as a function of Δμ/ε in the vicinity of the prewetting transition.

iii.2 SKA for a planar wall

For the sake of clarity and completeness we briefly review the main
features of SKA for a planar geometry (details are given in
Ref. Dietrich (1988)).

Let us consider a liquid film of a thickness ℓ adsorbed on a
planar wall. According to the SKA the density distribution is
approximated by a piecewise constant function

ρSKAℓ(z)=⎧⎪⎨⎪⎩0z<δ,ρ+lδ<z<ℓ,ρgz>ℓ,

(20)

where ρg is the density of the gas reservoir and ρ+l is the density of the metastable liquid at the same thermodynamic conditions stabilized by the presence of the planar wall, Eq.
(10) and δ≈12(σ+σw). The off-coexistence of the two phases induces the pressure difference

p+(μ)−p(μ)≈ΔρΔμ,

(21)

where p+ is the pressure of the metastable liquid and p is
the pressure of the gas reservoir, and where we assume that
Δμ=μ−μsat<0 is small.

The excess grand potential per unit area A of the system
then can be expressed in terms of macroscopic quantities as a
function of ℓ

Ωex(ℓ;μ)A

(22)

=−ΔμΔρ(ℓ−δ)+γSKAwl(μ)+γSKAlg+wSKA(ℓ;μ),

where γSKAwl and γSKAlg are the SKA to
the wall-liquid and the liquid-gas surface tensions, respectively,
and wSKA(ℓ) is the effective potential between the two
interfaces (binding potential). In the following, we will suppress
the explicit μ-dependence of these quantities.

The link with the microscopic theory can be made, if the
contributions in the right-hand-side of Eq. (22)
are expressed in terms of our molecular model, which, when summed
up, give the excess grand potential
(7) where we have substituted
the ansatz (20):

γSKAwl

=−ρ+2l2∫0−∞∫∞0ΦPla(|z−z′|)dz′dz+

(23)

+ρ+l∫∞δV∞(z)dz

=34πεσ4ρ+2l+π90δ8(σ6w−30δ6)σ6wρwεwρ+l.

γSKAlg

=−Δρ22∫0−∞∫∞0ΦPla(|z−z′|)dz′dz

=34πεσ4Δρ2

(24)

wSKA(ℓ)

=Δρ(ρ+l∫∞ℓ−δ∫∞zΦ{\text{Pla}}(z′)dz′dz−

−∫∞ℓV∞(z)dz)

(25)

=−A12πℓ2⎛⎜
⎜
⎜⎝1+2+3δℓ1−ρwεwσ6wρ+lεσ6δℓ+O((δ/ℓ)3)⎞⎟
⎟
⎟⎠,

where we considered the distinguished limit δ≪ℓ. A
is the Hamaker constant given by:

A=4π2Δρ(ρ+lεσ6−ρwεwσ6w).

(26)

We note that the Hamaker constant is implicitly temperature
dependent and that the attractive contribution of the potential of
the wall enables the Hamaker constant to change its sign. Hence, in
contrast with the adsorption on a hard wall, where the Hamaker
constant is always negative, there may be a temperature below which
its sign is positive (large ρl) and negative above. Clearly,
complete wetting is only possible for A<0.

Making use of only the leading-order term in
(25) the minimization of
(22) with respect to ℓ gives:

ΔρΔμ−A6πℓ3≈0.

(27)

Hence, at this level of approximation the equilibrium thickness of
the liquid film is:

ℓeq≈(A6πΔρΔμ)1/3.

(28)

When substituted into (22), the wall-gas surface
tension to leading order reads:

γSKAwg

=γSKAwl+γSKAlg+(−9A16π)1/3|ΔρΔμ|2/3.

(29)

Equation (28) can be confirmed by a comparison
against the numerical DFT, see Fig. 4.
We observe that the prediction of SKA becomes reliable for
|Δμ|<0.01ε corresponding to a somewhat
surprisingly small value of the liquid film, ℓ≈5σ.
Beyond this value, the coarse-grained approach looses its validity
and also the prewetting transition is approached, both of which
cause the curve in Fig. 4 to bend (see
also Fig. 3). It is worth noting that the
only term in (22) having an ℓ-dependence and
thus governing the wetting behavior, is the term related to the
undersaturation pressure and the binding potential,
wSKA(ℓ). Clearly, γlg does not come into play in
the planar case since the translation of the liquid-gas interface
along the z axis does not change the free energy of the system.
The situation becomes qualitatively different if the substrate is
curved. Nevertheless, at this stage we conclude in line with
earlier studies, that SKA provides a fully satisfactory approach to
the first-order wetting transition on a planar wall.

Figure 4: Log-log plot of the film thickness as a function of
deviation of the chemical potential from saturation, Δμ,
for kBT=0.7ε and wall parameters
ρwεw=0.8ε/σ3, σw=1.25σ. The crosses are results from DFT computations. The solid
line is the analytical prediction in
Eq. (27) obtained from SKA.

where ~R=R+δ. Within this approximation, the
liquid-vapour surface tension becomes (see also Appendix
B)

γSKAlg(R)=γSKAlg(∞)

[1−29ln(R/σ)(R/σ)2+O((σ/R)2)]

(32)

and an analogous expansion holds for γSKAwl(R).
The ln(R/σ)(R/σ)2 correction to γSKAlg(∞) is due to the r−6 decay of our
model. We note that short-range potentials lead to different
curvature dependence of the surface tension, a point that has been
discussed in detail in Refs.
Stewart and Evans (2005a); Evans et al. (2004); Parry et al. (2006). Interestingly, the
O(σ/R) correction to the surface tension, as one would expect
from the Tolman theory Tolman (1948), is missing. It corresponds to
a vanishing Tolman length within SKA, as we will explicitly show in
the following section.
Although the
value of the Tolman length is still a subject of some controversy, it is
most likely that its value is non-zero, unless the system is
symmetric under interchange between the two coexisting
phases Fisher and Wortis (1984). This observation has been confirmed
numerically in Ref. Stewart and Evans (2005a) from a fit of DFT results
for the wall-gas surface tension in a non-drying regime for the
hard-wall substrate. Thus, the linear term was included by hand into
the expansion (32) Stewart and Evans (2005a).

Finally, the binding potential within the SKA for the spherical wall yields

wSKA(ℓ;R)=wSKA(ℓ;∞)(1+ℓR)

(33)

where terms O((δ/ℓ)3,δ/R,ln(ℓ/R)(R/ℓ)2) have been neglected.

iv.2 SIA for the spherical wall

Figure 5: Sketch of the density profile according to SIA for a
certain film thickness ℓ. A piecewise function approximation is
employed so that except for the interval
(R+ℓ−χ/2,R+ℓ+χ/2) the density is assumed to be
piecewise constant.

As an alternative to SKA, Napiórkowski and
Dietrich Napiórkowski and Dietrich (1986) proposed a
modified version of the effective Hamiltonian, in which the
liquid-gas interface was approximated in a less crude way by a
continuous monotonic function, the SIA. Applied for the second-order
wetting transition on a planar wall, SIA merely confirmed that SKA
provides a reliable prediction for such a system. Formulated now for the spherical case, the
density profile of the fluid takes the form:

Thus, a non-zero width of the liquid-vapour interface, χ, is
introduced as an additional parameter. The density profile
ρlg(⋅) in this region is not specified, but the
following constraints are imposed:

ρlg(−χ2)=ρ+l %
and ρlg(χ2)=ρg,

(35)

with an additional assumption of a monotonic behaviour of the
function ρlg(r). An illustrative example of
ρSIAR,ℓ(r) is given in Fig. 5.
The corresponding excess grand potential takes the form

Ωex4πR2=

−ΔμΔρ(R+ℓ)3−~R33R2+γSIAwl(R)+

+(1+ℓR)2γSIAlg(R+ℓ)+wSIA(R,ℓ),

(36)

taking R+ℓ as the Gibbs dividing surface
(so that ℓ is a measure of the number of
particles adsorbed at the wall). The binding potential
(see also Appendix C.3) is obtained from

wSIA

(R,ℓ)=

=

ρ+l∫∞R+ℓ−χ/2(ρ+l−ρSIAR,ℓ(r))ΨR+δ(r)(rR)2dr−

−

∫∞R+ℓ−χ/2(ρ+l−ρSIAR,ℓ(r))VR(r)(rR)2dr,

(37)

where ΨR(r)=∫R0ΦSph(r,r′)dr′ – see
Appendix B.1 for the explicit form
of the last expression.

From now on, we neglect the curvature dependence of χ and
ρlg,R(⋅), as they would introduce higher-order
corrections not affecting the asymptotic results at our level of
approximation. This is also in line with previous studies which show
that the Tolman length only depends on the density profile in the
planar limit Fisher and Wortis (1984). Then (38) can be
written as

γSIAlg(R)

=γSIAlg(∞)[1−2δ∞R+O(ln(R/σ)(R/σ)2)],

(39)

where δ∞ is the Tolman length of the liquid-gas surface
tension, as given by (Appendix C.2):

δ∞=1γSIAlg(∞)∫χ/2−χ/2(p(ρlg(z))−pref)zdz.

(40)

The Tolman length is independent of the choice of the dividing surface.
We also note that an
immediate consequence of Eq. (40) is that
within SKA the Tolman length vanishes.

The equilibrium film thickness then follows from setting the derivative
of (36)
w.r.t. ℓ equal to zero:

with fI(r)=ρ+lΨR+δ(r)(rR)2
and fII(r)=VR(r)(rR)2. Since
ρlg(r) is monotonic, i.e. ρ′lg does not change sign,
the mean value theorem can be employed such that

∫χ/2−χ/2ρ′lg(r)fI,II(R+ℓ

+r)dr=

(43)

−ΔρfI,II(R+ℓ+ξI,II),

for some ξI,II∈(−χ/2,χ/2), where we made use
of ∫ρ′lg(r)dr=−Δρ.
Substituting (43) into
(41) and setting
the resulting expression equal to zero, we obtain:

Δμ=

1Δρ⎛⎜⎝2γSIAlg(R+ℓ)R+ℓ+dγSIAlgdℓ∣∣
∣∣R+ℓ⎞⎟⎠−

−ρ+lΨR+δ(R+ℓ+ξI)(1+ξIR+ℓ)2+

+VR(R+ℓ+ξII)(1+ξIIR+ℓ)2.

(44)

So far, there is no approximation within SIA. Equation
(44) can be simplified by
appropriately estimating the values of the auxiliary parameters
ξI and ξII. To this end, we employ a simple linear
approximation to the density profile at the liquid-gas interface,
taking −ρ′lg(r)/Δρ≈1/χ in
(43). Furthermore, we expand fI,II in powers
of ℓ/R,σ/ℓ

fI(R+ℓ+r)=

−2πρ+lεσ63(ℓ+r−δ)3(1+ℓ+r+3δ2R+

+O((σℓ)6,(ℓR)2)),

(45)

fII(R+ℓ+r)=

−2πρwεwσ6w3(ℓ+r)3(1+ℓ+r2R+

(46)

+O((σℓ)6,(ℓR)2)),

where we assumed the distinguished limits r,δ,σ≪ℓ≪R. Inserting (45) and (46)
into (43) yields for ξi:

Finally, substituting (48) and (49) into
(44) we have to leading order:

ΔρΔμ−2RγSIAlg(∞)≈

A6πℓ3,

(50)

and hence, to leading order the equilibrium wetting film thickness
is:

ℓSIAeq≈⎛⎜
⎜⎝A6π(ΔρΔμ−2γSIAlg,∞/R)⎞⎟
⎟⎠1/3.

(51)

We note that this asymptotic analysis can be extended beyond
(51), by including terms O(δ/ℓ), O(ℓ/R) and
O((χ/ℓ)2). The latter occurs due to the “soft”
treatment of the liquid-vapor interface and is thus not present in
SKA.

In Fig. 6 we compare two
adsorption isotherms (kBT=0.7ε) corresponding to
wetting on a planar and a spherical wall (R=100σ). The two
curves are mutually horizontally shifted by a practically constant
value, in accordance with Eq. (50). This
implies that the curve for the spherical wall crosses the saturation
line Δμ=0 at a finite value of ℓ, and eventually
converges to the saturation line as Δμ−1 from the right,
thus the finite curvature prevents complete wetting. The horizontal
shift corresponds to the Laplace pressure contribution,
Δμ=2γSIAlg(∞)/(ΔρR), as
verified by comparison with the numerical DFT,
Fig. 7. All these conclusions are in line with SKA.
However, the difference between SKA and SIA consists in a different
treatment of γlg(∞), compare
(68) and
(81). This is quite obvious, since the
softness of the interface influences the free energy required to
increase the film thickness. We will discuss this point in more
detail in the following section.

Figure 6: Isotherms and density profiles for a planar wall (dashed
lines) and a sphere with R=100σ (solid lines) at kBT=0.7ε and with wall parameters, ρwεw=0.8ε/σ3 and σw=1.25σ. To directly
compare the planar to the spherical case, the film thickness instead
of adsorption is used as a measure. The subplots in the inset depict
the density ρσ3 as a function of the distance from the
wall z/σ and (r−R)/σ for the planar and the spherical
cases, respectively. The points A and A′ are at the prewetting
transitions. Points B,B′ and C,C′ correspond to the same film
thickness. B is at saturation whereas C is chosen such that the
film thickness ℓ is 20σ.
Figure 7: Numerical verification of
Eq. (50). The film thickness ℓ is
fixed and corresponds to the adsorption ΓR=3.905/σ2. The solid line corresponds to the analytical
result, Δμ−2γSIAlg(∞)/(ΔρR)=Cε, where γSIAlg(∞)=0.524ε/σ2, see Table 1. The
symbols denote the numerical DFT results.

iv.3 Comparison of SKA and SIA

We now examine the repercussions of the way the liquid-gas interface
is treated on the prediction of wetting behaviour on a spherical
surface.
As already mentioned in Sec. IV.2, the linear
correction in the curvature to the planar liquid-gas surface
tension, ignored within SKA, is properly captured by SIA.
Furthermore, the presence of the Laplace pressure suggests that the
liquid-gas surface tension plays a strong part in the determination
of the equilibrium film thickness. This contrasts to the case of a
planar geometry, where the term associated with the liquid-gas
surface tension has no impact on the equilibrium configuration.

To investigate this point in detail, we will first compare the
approximations of γlg as obtained by the two approaches.
For this purpose, we start with SIA for a given parameterization of
the liquid-gas interface. As shown in Table 1,
we employ linear, cubic and hyperbolic tangent auxiliary functions,
where the latter violates condition (35)
negligibly. The particular parameters are determined by minimization
of a given function with respect to the corresponding parameters. In
Table 1 we display the planar liquid-gas
surface tension associated with a particular parameterization and
the Tolman length resulting from Eq. (40) for
the temperature kBT=0.7ε. In all three cases the
surface tension is close to the one obtained from the numerical
solution of DFT and also the predictions of the Tolman length are in
a reasonable agreement with the most recent simulation
results Sampayo et al. (2010); Block et al. (2010); van Giessen and Blokhuis (2009), with thermodynamic results Bartell (2001) as well as
with results from the van der Waals square gradient theory Blokhuis and Kuipers (2006).

Auxiliary function ρlg(z)

γSIAlg(∞)

argument

δ∞

¯ρ−Δρzχ

0.544ε/σ2

χ=4.0σ

−0.07σ

¯ρ−32Δρzχ+2Δρ(zχ)3

0.532ε/σ2

χ=5.4σ

−0.09σ

¯ρ−Δρ2tanh(αz/σ)

0.524ε/σ2

α=0.66

−0.11σ

Table 1: Planar surface tensions (81), Tolman lengths (40) and the
corresponding parameters for temperature kBT=0.7ε
according to a given auxiliary function approximating the density
distribution of the vapour-liquid interface. The parameters are from
auxiliary function minimization. The surface tension given by
numerical DFT computations is γlg=0.517ε/σ2 and ¯ρ=(ρl+ρg)/2. Note that
in the tanh-case, the interface width is implicitly determined by the steepness
parameter α.Figure 8: Plot of a dimensionless planar liquid-gas surface tension
for the liquid-gas interface approximation ρ(z)=ρl+ρg2−Δρ2tanh(αz/σ) for kBT=0.7ε as a function of the
steepness parameter α. The upper dashed line is the surface
tension obtained from SKA, whereas the lower dashed line displays
the surface tension obtained from numerical DFT.

It is reasonable to assume that from the set of considered
auxiliary functions, the tanh-approximation is the most realistic
one, although the numerical results as given in Table
1 suggest that it is mainly the finite width
of the liquid-gas interface, rather than the approximation of the
density profile at this region, that matters. To illustrate this, we
show in Fig. 8 the dependence of the
surface tension on the steepness parameter α, determining the
shape of the tanh function. Note that the limit α→∞ corresponds to the surface tension as predicted by SKA,
γSKAlg,∞=1.060ε/σ2, for
kBT=0.7ε. Such a value contrasts with the result of
SIA, which corresponds to the minimum of the function, and yields
γSIAlg,∞=0.524ε/σ2, in much
better agreement with the numerical solution of DFT,
γDFTlg,∞=0.517ε/σ2.

Asymptotic analysis of the film thickness in Eq.
(50), reveals that the film thickness for
large but finite R remains finite even at saturation with
ℓ∼R13 in line with earlier studies, e.g.
Refs. Bieker and Dietrich (1998); Stewart and Evans (2005a). From
Eq. (50) one also recognizes a strong
dependence of ℓ on the planar liquid-gas surface tension. In
Fig. 9 we present the SIA and SKA
predictions of the dependence on ℓ as a function of the wall
radius. The comparison with the numerical DFT results reveals that
for large R SIA is clearly superior, reflecting a more realistic
estimation of the liquid-gas surface tension.
For small values of R (and ℓ) we observe a deviation between DFT and the SIA results.
This indicates a limit of validity of our first-order analysis and the assumption of large film thicknesses.

Figure 9: Film thickness at saturation (Δμ=0) as a
function of the wall radius. The symbols correspond to the numerical
DFT results. The dashed line shows the prediction according to
Eq. (51), where γSIAlg(∞)=0.524ε/σ2 (see Table 1). The
dash-dotted line corresponds to Eq. (51) where
γSKAlg(∞)=1.060ε/σ2 is used
instead of γSIAlg(∞).
The wall parameters are ρwεw=0.8ε/σ3 and σw=1.25σ at kBT=0.7ε.Figure 10: Density profiles of the fluid adsorbed at the spherical walls of radii R=104.1σ (dashed) and R=210.6σ (dashed-dotted)
in a saturated state and at the planar wall (solid line) in an undersaturated state, Δμ=−0.015ε .
The wall radii correspond to the equality 2γjlg,∞/R=Δρ|Δμ| for j=SIA (dashed) and j=SKA (dashed-dotted).
For kBT=0.7ε.
The wall
parameters are ρwεw=0.8ε/σ3 and
σw=1.2σ.